Fixed-time sliding-mode formation tracking control method for multiple ocean surface ships

Abstract

The invention relates to the technical field of tracking control, and particularly provides a fixed-time sliding mode formation tracking control method for multiple ocean surface ships. The method comprises the following steps: constructing a dynamic model of a marine surface ship MSV; designing a disturbance observer with fixed time according to the dynamic model of the MSV; according to the method, a fixed-time sliding mode formation controller is designed through a fixed-time disturbance observer and an improved sliding mode reaching law, and through the fixed-time disturbance observer, the anti-disturbance capability of a plurality of ocean surface ships is improved; the control precision is improved through the sliding mode formation controller with the fixed time, the convergence speed is improved through the improved sliding mode reaching law, and the chattering phenomenon caused by sliding mode control is relieved.

Description

Technical Field

[0001] The present invention relates to the field of tracking control technology, and in particular to a method for tracking control of a fixed-time sliding mode formation of multiple ocean surface ships. Background Art

[0002] In recent decades, marine surface vessels (MSVs) have attracted increasing attention due to their wide range of applications in many fields, including marine surveys, maritime rescue, and surveillance. However, it is often challenging for a single MSV to complete certain specialized missions, such as on-board replenishment, seafloor mapping, and ocean patrols. To overcome the limitations associated with single-vessel operations, scholars have begun to focus on the formation control of multiple MSVs, which has now become a hot topic.

[0003] A variety of control methods have been proposed to achieve formation control of multiple MSVs, such as adaptive control, neural networks, and sliding mode control. Early results on formation control of multiple MSVs focused on the asymptotic convergence of the tracking error, emphasizing the steady-state performance of the system over infinite time intervals. To enhance the transient performance of the system, finite-time control methods have been employed in recent years for formation control of multiple MSVs. For example, a distributed finite-time controller for multiple MSVs is designed using sliding mode control to achieve finite-time formation. Furthermore, to overcome input saturation constraints and the effects of disturbances, a finite-time extended state observer-based formation control method for multiple MSVs is proposed. Within the framework of finite-time controller design, the convergence time of the formation tracking error often depends on the initial conditions of the system, which may not be available in advance. To overcome this limitation, fixed-time control methods have been employed to solve the formation control problem of multiple MSVs, and significant progress has been made.

[0004] Furthermore, the inherent complex uncertainties of ship dynamics models, coupled with potential disturbances from marine environmental factors such as wind, waves, and currents, have led to significant attention in designing robust formation control strategies for multiple MSVs. For example, a fixed-time output feedback control scheme combined with an extended state observer has been proposed for trajectory tracking control of MSVs subject to unknown disturbances and uncertainties. Furthermore, disturbance estimation and rejection are implemented, and tracking performance is improved by designing a nonsingular terminal sliding mode controller based on a fixed-time disturbance observer. Furthermore, actuator failures are often unavoidable in the actual operation of MSVs. Several fault-tolerant control methods have been proposed to address this issue. Despite significant progress in solving the formation control problem for multiple MSVs, research on fixed-time formation tracking control for MSVs with mixed effects (external disturbances, uncertainties, input saturation constraints, and actuator failures) is insufficient. Furthermore, the state-of-the-art nonsingular terminal sliding mode formation control methods still have room for improvement in terms of response speed and flutter amplitude. Summary of the Invention

[0005] In view of this, the present invention provides a method for fixed-time sliding mode formation tracking control of multiple ocean surface ships, which is used to improve the anti-disturbance capability of multiple ocean surface ships, improve control accuracy, increase convergence speed, and reduce vibration.

[0006] In a first aspect, the present invention provides a method for tracking and controlling a fixed-time sliding mode formation of multiple ocean surface ships, the method comprising:

[0007] Step 1: Construct a dynamic model of the ocean surface ship MSV;

[0008] Step 2: Design a fixed-time disturbance observer based on the dynamic model of MSV;

[0009] Step 3: Design a fixed-time sliding mode formation controller SMFC through a fixed-time disturbance observer and an improved sliding mode reaching law.

[0010] Optionally, step 1 includes:

[0011] Assuming there is a group of ocean surface ships MSV, the dynamic model of ocean surface ship i is expressed as:

[0012]

[0013] Among them, i=1,2...n, η i =(x i ,y i ,ψ i ) T and v i =(u i ,v i ,r i ) T Respectively represent the position and velocity vectors in the fixed earth coordinate system; M i ∈R 3 , C i (v i )∈R 3 and D i (v i )∈R 3 Represent three dynamic matrices respectively; τ wi =(τ wiu ,τ wiv ,τ wir ) T and τ i =(τ iu ,τ iv ,τ ir ) TRepresent external disturbance and input respectively; rotation matrix R i (ψ i ) is expressed as:

[0014]

[0015]

[0016]

[0017] Considering the problems of actuator failure and input saturation constraints, the control input τ i The definition is as follows:

[0018] τ i =h(τ A )+[(Z(t)-I)h(τ A )+τ ib ]=h(τ A )+τ fi (5)

[0019] Among them, τ A represents the actual control input, τ ib represents an additional bias fault; Z(t) = diag{z1,z2,z3} represents the coefficient matrix of the actuator; if z1 = 1, z2 = 1, z3 = 1 and τ ib =0, then the i-th actuator has no fault; 0 <z i <1(i=1,2,3) and τ ib ≠0 means the actuator will encounter failure and saturation problems; h(τ A )=sgn(τ A )min(|τ A |,τ AM ) represents the saturation function, where τ AM is a constant related to the actuator limit;

[0020] Through formula (1) to formula (5), the dynamic system is obtained as follows:

[0021]

[0022] Among them, M aci =S i (ψ i )M i S i (ψ i ) T , C aci =S i (ψ i )(C i (v i )-M iA(r))S i (ψ i ) T , τ aci =S i (ψ i )h(τ A ), D aci =S i (ψ i )D i (v i )S i (ψ i ) T ;

[0023] The matrix of the system is defined as follows:

[0024]

[0025] in, Represent the standard parameter matrix, M0, C0, D0 represent the unknown parameter matrix; the dynamic system is further obtained as:

[0026]

[0027] in, Indicates that the system is complex and uncertain;

[0028] The expected trajectory is defined as:

[0029]

[0030] Among them, η d ,v d ,u d represents the position, velocity vector and control input of the leading MSV;

[0031] Position and velocity tracking will be achieved, denoted as η i →η d and v i →v d ; Define relative formation shape h i ∈R 3 , then the formation tracking error is designed as:

[0032] e 1i =η i -h i -η d (9)

[0033]

[0034] Among them, e 1i =[x e1i ,y e1i,ψ e1i ] represents position error;

[0035] Therefore, formula (1) can be rewritten as:

[0036]

[0037]

[0038] in, and denote the conversion control input and lumped uncertainty, respectively.

[0039] Optionally, step 2 includes:

[0040] The uncertainty of MSV is considered as a composite perturbation d i , the composite disturbance is estimated by designing a fixed-time disturbance observer;

[0041] The disturbance observer is designed as:

[0042]

[0043] in, Indicates d i Estimates of z 1i ∈R 3 represents an auxiliary state vector, z 2i represents output, γ1>0, γ2>0 represent constant parameters;

[0044] Since the auxiliary state cannot be measured, z 1i The estimated design is:

[0045]

[0046] in, represents the estimation error of the auxiliary state, and γ3>0,λ1>0,λ2>0,0 <k1<1,k2> 1;

[0047] Assumption 1: Lumped uncertainty d i satisfy is a bounded constant;

[0048] Lemma 1: Consider a Lyapunov function V(x) that is defined in a neighborhood D of the origin and satisfies Where λ1,λ2>0,0<α<1,β≥1, then the origin of the MSVs is fixed-time stable; therefore, the residence time satisfies

[0049] Lemma 2: If Assumption 1 holds, and the parameters satisfy γ3>0,λ1>0,λ2>0,0 <k1<1,k2> 1, then the observation error converges to 0 within a fixed time T0, satisfying:

[0050]

[0051] Optionally, first, show that the estimated error of the auxiliary state converges to 0 in constant time; then, prove the constant-time convergence of the estimated error of the disturbance observer;

[0052] Step 21. Consider the Lyapunov function as:

[0053]

[0054] The derivative function of Lyapunov is:

[0055]

[0056] According to Lemma 1, the auxiliary state estimation error converges to 0, and the fixed time is obtained:

[0057]

[0058] Step 22: Define the disturbance observation error as According to formula (20), we get:

[0059]

[0060] Therefore, combining formula (20) and formula (26), the observation error is further obtained:

[0061]

[0062] Therefore, the perturbation observation error will converge to 0 at t>T0, then the fixed-time disturbance observer can accurately estimate the composite uncertainty.

[0063] Optionally, step 3 includes:

[0064] Design a fixed-time non-singular sliding mode surface, which is expressed as:

[0065]

[0066]

[0067]

[0068] The derivative of a non-singular sliding surface is expressed as:

[0069]

[0070] Combining the fixed-time disturbance observer, the non-singular sliding mode surface and the improved sliding mode reaching law, the fixed-time sliding mode formation controller SMFC is designed as follows:

[0071] u i =u i1 +u i2 (31)

[0072]

[0073]

[0074]

[0075] Optionally include:

[0076] Theorem 1: If Assumption 1 holds, and the gain satisfies

[0077] Non-singular sliding mode control can achieve fixed-time formation tracking, and the stability time T is bounded by:

[0078] T≤max(T1,T2)+T0 (35)

[0079] Proof: The proof process is divided into two parts:

[0080] Part 1:

[0081]

[0082] Based on the formula = and u i =u i1 +u i2 , the derivative of the non-singular sliding mode surface can be rewritten as:

[0083]

[0084] The Lyapunov function is chosen as:

[0085]

[0086] The derivative of the Lyapunov function is:

[0087]

[0088] Due to the characteristics of the saturation function, two cases are considered:

[0089] Case 1: ||S i ||>φ;

[0090]

[0091] Case 2: ||S i ||<φ;

[0092]

[0093] In summary, we get:

[0094]

[0095] According to Lemma 1, the fixed time of the reaching capacity of the sliding mode reaching law is obtained as follows:

[0096]

[0097] Part 2:

[0098] When S i When it converges to 0, based on the above results, the convergence of the tracking error will be proved; in order to evaluate the stability of the system, the Lyapunov function is selected as

[0099]

[0100] The derivative of the Lyapunov function is:

[0101]

[0102] According to Lemma 1, the tracking error will converge to 0 within a fixed time T2, which is:

[0103]

[0104] According to part 2, the tracking error converges to 0 after T2; considering part 1, the total convergence time is limited by T3, satisfying T3≤max(T1,T2)+T0; according to If e 1i can converge to 0, then e 2i It will converge to 0 within a fixed time, that is, the fixed-time sliding mode formation controller will achieve formation tracking.

[0105] Optionally, the improved sliding mode reaching law includes:

[0106] The sliding mode reaching law is improved based on the saturation function and exponential function as follows:

[0107]

[0108]

[0109] Among them, α1>0, α2>0, α3=k / Ω(Si ),Ω(S i ) = η + γ(1 + |S i | / μ) -1 ,k > 0,η > 0,0 < γ < 1,μ = α|S0|,0 < Φ = φ c < 1,0 < φ < 1, is an odd function; in addition, a, b, c are three parameters of the improved sliding mode reaching law, satisfying 0 < c < a < 1 and 0 < b < 1;

[0110] When the system state is far from the sliding mode surface, |S i | > φ, which means the corresponding sliding mode reaching law is:

[0111] When the system state is close to the sliding mode surface, gradually becomes smaller, playing a role in reducing chattering;

[0112] When the system state gradually approaches the sliding mode surface, |S i | ≤ φ, which means the corresponding reaching law is expressed as:

[0113] In a second aspect, an embodiment of the present invention provides a computer-readable storage medium, where the computer-readable storage medium includes a stored program, and when the program runs, it controls the device where the computer-readable storage medium is located to execute the method for fixed-time sliding mode formation tracking control of multiple ocean surface ships in the first aspect or any possible implementation manner of the first aspect.

[0114] In a third aspect, an embodiment of the present invention provides an electronic device, including: one or more processors; a memory; and one or more computer programs, where the one or more computer programs are stored in the memory, and the one or more computer programs include instructions, and when the instructions are executed by the device, the device is made to execute the method for fixed-time sliding mode formation tracking control of multiple ocean surface ships in the first aspect or any possible implementation manner of the first aspect.

[0115] In the technical solution provided by the present invention, the method includes constructing a dynamic model of an ocean surface ship MSV; designing a fixed-time disturbance observer according to the dynamic model of MSV; designing a fixed-time sliding mode formation controller through the fixed-time disturbance observer and the improved sliding mode reaching law. This method improves the anti-disturbance ability of multiple ocean surface ships through the fixed-time disturbance observer; improves the control accuracy through the fixed-time sliding mode formation controller, and improves the convergence speed and reduces the chattering phenomenon caused by sliding mode control through the improved sliding mode reaching law. BRIEF DESCRIPTION OF THE DRAWINGS

[0116] In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the following briefly introduces the drawings required for use in the embodiments. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative work.

[0117] Figure 1 A flow chart of a method for fixed-time sliding mode formation tracking control of multiple ocean surface ships provided in an embodiment of the present invention;

[0118] Figure 2 A schematic diagram of a formation tracking trajectory provided by an embodiment of the present invention;

[0119] Figure 3 A schematic diagram of tracking error provided by an embodiment of the present invention; Figure 3 (a) is a schematic diagram of the tracking error of x; Figure 3 (b) Schematic diagram showing the tracking error of y; Figure 3 (c) Schematic diagram showing the tracking error of ψ;

[0120] Figure 4 The lumped disturbance d provided in the embodiment of the present invention i and fixed-time observations Schematic diagram for comparison; Figure 4 (a) is d iu Schematic diagram of comparison; Figure 4 (b) is d iv Schematic diagram of comparison; Figure 4 (c) is d ir Schematic diagram of comparison;

[0121] Figure 5 The formation controller τ provided by the embodiment of the present invention i Schematic diagram of; Figure 5 (a) is τ iu Schematic diagram of; Figure 5 (b) is τ iv Schematic diagram of; Figure 5 (c) is τ ir Schematic diagram of;

[0122] Figure 6 A schematic diagram of the x comparison error provided by an embodiment of the present invention;

[0123] Figure 7 A schematic diagram of the y comparison error provided by an embodiment of the present invention;

[0124] Figure 8 A schematic diagram of a control input comparison of MSV1 provided in an embodiment of the present invention;

[0125] Figure 9 A schematic diagram of control input comparison of MSV2 provided in an embodiment of the present invention;

[0126] Figure 10 A schematic diagram of control input comparison of the MSV3 provided in an embodiment of the present invention;

[0127] Figure 11 A schematic diagram of control input comparison of the MSV4 provided in an embodiment of the present invention;

[0128] Figure 12 A schematic diagram of an electronic device provided by an embodiment of the present invention. DETAILED DESCRIPTION

[0129] To make the objectives, technical solutions, and advantages of the embodiments of the present invention more clear, the technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative efforts shall fall within the scope of protection of the present invention.

[0130] It should be understood that the embodiments described are only a portion of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by persons of ordinary skill in the art without creative work are within the scope of protection of the present invention.

[0131] The terms used in the embodiments of the present invention are only for the purpose of describing specific embodiments and are not intended to limit the present invention. The singular forms "a", "the" and "the" used in the embodiments of the present invention are also intended to include plural forms, unless the context clearly indicates other meanings.

[0132] It should be understood that the term "and / or" as used herein simply describes a relationship between associated objects, indicating that three possible relationships exist. For example, "A and / or B" can represent: A alone, A and B together, or B alone. Furthermore, the character " / " in this document generally indicates an "or" relationship between the associated objects.

[0133] The word "if," as used herein, may be interpreted as "at the time of" or "when" or "in response to determining" or "in response to detecting," depending on the context. Similarly, the phrases "if it is determined" or "if (stated condition or event) is detected" may be interpreted as "when it is determined" or "in response to the determination" or "when detecting (stated condition or event)" or "in response to detecting (stated condition or event)," depending on the context.

[0134] Figure 1 The flow chart of the method for fixed-time sliding mode formation tracking control of multiple ocean surface ships provided by the embodiment of the present invention is as follows: Figure 1 As shown, the method includes:

[0135] Step 1: Construct a dynamic model of the ocean surface ship MSV.

[0136] In an embodiment of the present invention, step 1 includes:

[0137] Assuming there is a group of ocean surface ships MSV, the dynamic model of ocean surface ship i is expressed as:

[0138]

[0139] Among them, i=1,2...n, η i =(x i ,y i ,ψ i ) T and v i =(u i ,v i ,r i ) T Respectively represent the position and velocity vectors in the fixed earth coordinate system; M i ∈R 3 , C i (v i )∈R 3 and D i (v i )∈R 3 Represent three dynamic matrices respectively; τ wi =(τ wiu ,τ wiv ,τ wir ) T and τ i =(τ iu ,τ iv ,τ ir ) T Represent external disturbance and input respectively; rotation matrix R i (ψ i ) is expressed as:

[0140]

[0141]

[0142]

[0143] Considering the problems of actuator failure and input saturation constraints, the control input τ i The definition is as follows:

[0144] τ i =h(τ A )+[(Z(t)-I)h(τ A )+τ ib ]=h(τ A )+τ fi (5)

[0145] Among them, τ A represents the actual control input, τ ib represents an additional bias fault; Z(t) = diag{z1,z2,z3} represents the coefficient matrix of the actuator; if z1 = 1, z2 = 1, z3 = 1 and τ ib =0, then the i-th actuator has no fault; 0 <z i <1(i=1,2,3) and τ ib ≠0 means the actuator will encounter failure and saturation problems; h(τ A )=sgn(τ A )min(|τ A |,τ AM ) represents the saturation function, where τ AM is a constant related to the actuator limit;

[0146] Through formula (1) to formula (5), the dynamic system is obtained as follows:

[0147]

[0148] Among them, M aci =S i (ψ i )M i S i (ψ i ) T , C aci =S i (ψ i )(C i (v i )-M i A(r))S i (ψ i ) T , τ aci =S i (ψ i )h(τ A ), D aci =S i (ψ i )D i (v i )S i (ψ i ) T ;

[0149] The matrix of the system is defined as follows:

[0150]

[0151] in, Represent the standard parameter matrix, M0, C0, D0 represent the unknown parameter matrix; the dynamic system is further obtained as:

[0152]

[0153] in, Indicates that the system is complex and uncertain;

[0154] The expected trajectory is defined as:

[0155]

[0156] Among them, η d ,v d ,u d represents the position, velocity vector and control input of the leading MSV;

[0157] The present invention aims to solve the problem of fixed-time formation tracking of multiple MSVs subject to external disturbances, model uncertainty, actuator failures, and input saturation constraints. So far, position and velocity tracking is achieved, denoted as η i →η d and v i →v d ; Define relative formation shape h i ∈R 3 , then the formation tracking error is designed as:

[0158] e 1i =η i -h i -η d (9)

[0159]

[0160] Among them, e 1i =[x e1i ,y e1i ,ψ e1i ] represents position error;

[0161] Therefore, formula (1) can be rewritten as:

[0162]

[0163]

[0164] in, and denote the conversion control input and lumped uncertainty, respectively.

[0165] Step 2: Based on the dynamic model of MSV, design a fixed-time disturbance observer.

[0166] In an embodiment of the present invention, step 2 includes:

[0167] In the marine environment, MSV may be affected by external disturbances, actuator failures, input saturation constraints and model uncertainties. The uncertainty of MSV is considered as a composite disturbance d i , the composite disturbance is estimated by designing a fixed-time disturbance observer;

[0168] The disturbance observer is designed as:

[0169]

[0170] in, Indicates d i Estimates of z 1i ∈R 3 represents an auxiliary state vector, z 2i represents output, γ1>0, γ2>0 represent constant parameters;

[0171] Since the auxiliary state cannot be measured, z 1i The estimated design is:

[0172]

[0173] in, represents the estimation error of the auxiliary state, and γ3>0,λ1>0,λ2>0,0 <k1<1,k2> 1;

[0174] Assumption 1: Lumped uncertainty d i satisfy is a bounded constant;

[0175] Lemma 1: Consider a Lyapunov function V(x) that is defined in a neighborhood D of the origin and satisfies Where λ1,λ2>0,0<α<1,β≥1, then the origin of the MSVs is fixed-time stable; therefore, the residence time satisfies

[0176] Lemma 2: If Assumption 1 holds, and the parameters satisfy γ3>0,λ1>0,λ2>0,0 <k1<1,k2> 1, then the observation error converges to 0 within a fixed time T0, satisfying:

[0177]

[0178] In an embodiment of the present invention, the present invention includes:

[0179] First, we show that the estimation error of the auxiliary state converges to 0 in constant time. Then, we prove the constant-time convergence of the estimation error of the disturbance observer.

[0180] Step 21. Consider the Lyapunov function as:

[0181]

[0182] The derivative function of Lyapunov is:

[0183]

[0184] According to Lemma 1, the auxiliary state estimation error converges to 0, and the fixed time is:

[0185]

[0186] Step 22: Define the disturbance observation error as According to formula (20), we get:

[0187]

[0188] Therefore, combining formula (20) and formula (26), the observation error is further obtained:

[0189]

[0190] Therefore, the perturbation observation error will converge to 0 at t>T0, then the fixed-time disturbance observer can accurately estimate the composite uncertainty.

[0191] Step 3: Design a fixed-time sliding mode formation controller SMFC through a fixed-time disturbance observer and an improved sliding mode reaching law.

[0192] In an embodiment of the present invention, step 3 includes:

[0193] Design a fixed-time non-singular sliding mode surface, which is expressed as:

[0194]

[0195]

[0196]

[0197] The derivative of a non-singular sliding surface is expressed as:

[0198]

[0199] Combining the fixed-time disturbance observer, the non-singular sliding mode surface and the improved sliding mode reaching law, the fixed-time sliding mode formation controller SMFC is designed as follows:

[0200] u i =u i1 +u i2 (31)

[0201]

[0202]

[0203]

[0204] In an embodiment of the present invention, the present invention includes:

[0205] Theorem 1: If Assumption 1 holds, and the gain satisfies

[0206] Non-singular sliding mode control can achieve fixed-time formation tracking, and the stability time T is bounded by:

[0207] T≤max(T1,T2)+T0 (35)

[0208] Proof: The proof process is divided into two parts:

[0209] Part 1:

[0210]

[0211] Based on the formula = and u i =u i1 +u i2 , the derivative of the non-singular sliding mode surface can be rewritten as:

[0212]

[0213] The Lyapunov function is chosen as:

[0214]

[0215] The derivative of the Lyapunov function is:

[0216]

[0217] Due to the characteristics of the saturation function, two cases are considered:

[0218] Case 1: ||S i ||>φ;

[0219]

[0220] Case 2: ||S i ||<φ;

[0221]

[0222] In summary, we get:

[0223]

[0224] According to Lemma 1, the fixed time of the reaching capacity of the sliding mode reaching law is obtained as follows:

[0225]

[0226] Part 2:

[0227] When S i When it converges to 0, based on the above results, the convergence of the tracking error will be proved; in order to evaluate the stability of the system, the Lyapunov function is selected as

[0228]

[0229] The derivative of the Lyapunov function is:

[0230]

[0231] According to Lemma 1, the tracking error will converge to 0 within a fixed time T2, which is:

[0232]

[0233] According to part 2, the tracking error converges to 0 after T2; considering part 1, the total convergence time is limited by T3, satisfying T3≤max(T1,T2)+T0; according to If e 1i can converge to 0, then e 2i It will converge to 0 within a fixed time, that is, the fixed-time sliding mode formation controller will achieve formation tracking.

[0234] The sliding mode formation controller proposed in the present invention enables the tracking error to converge to 0 within a fixed time; by using an improved sliding mode reaching law, the convergence speed is improved; at the same time, the method control can be applied to other fields.

[0235] In order to improve the dynamic performance of sliding mode control in the arrival phase, an improved sliding mode reaching law is designed. The traditional reaching law is expressed as:

[0236] Constant Reaching Law (CRL):

[0237]

[0238] Exponential Reaching Law (ERL):

[0239]

[0240] Power Reaching Law (PRL):

[0241]

[0242] Improved Double Power Reaching Law (IDPRL):

[0243]

[0244] Double Power Reaching Law (DPRL):

[0245]

[0246] Among them, γ1>0, γ2>0, a>0, λ1>0, λ2>0, S i is the sliding surface function, k1, k2 and S i Related, N(S i ) is S i function,

[0247] Due to the existence of the sign function in the reaching law, the controller may switch at high frequency. Therefore, it is necessary to solve this problem by replacing the sign function. The exponential function exp(·) provides a faster response.

[0248] In an embodiment of the present invention, the improved sliding mode reaching law includes:

[0249] The sliding mode reaching law is improved based on the saturation function and exponential function as follows:

[0250]

[0251]

[0252] Among them, α1>0, α2>0, α3=k / Ω(S i ),Ω(S i )=η+γ(1+|S i | / μ) -1 , k>0, η>0, 0<γ<1, μ=α|S0|, 0<Φ=φ c <1, 0<φ<1, is an odd function; in addition, a, b, and c are three parameters of the improved sliding mode reaching law, satisfying 0 < c < a < 1 and 0 < b < 1; the decision maker can flexibly set the values of the three parameters to adjust the approaching speed of the sliding mode control.

[0253] When the system state is far from the sliding mode surface, |S i | > φ, which means the corresponding sliding mode reaching law is: Note that

[0254] Therefore, compared with the traditional sliding mode reaching law, the sliding mode reaching law proposed by the present invention improves the response speed of the system. -α3S i The existence of ensures a further improvement in the convergence speed.

[0255] When the system state approaches the sliding mode surface, gradually becomes smaller, playing a role in reducing chattering;

[0256] When the system state gradually approaches the sliding mode surface, |S i | ≤ φ, which means the corresponding reaching law is expressed as: When |S i | → 0,

[0257] At the same time, the existence of -α3S i ensures that the amplitude can be reduced compared with the general gain.

[0258] In the embodiments of the present invention, the improved sliding mode reaching law is different from the latest sliding mode control method. By introducing a product term, the chattering amplitude is reduced and the response speed is improved. Therefore, compared with CRL and ERL, the sliding mode reaching law proposed by the present invention can eliminate oscillations. In addition, compared with PRL, DPRL, and IDPRL, although oscillations are also eliminated, the sliding mode reaching law proposed by the present invention has a faster approaching speed.

[0259] Simulation and comparison results: <>

[0260] Construct numerical results to prove the effectiveness of the designed control method:

[0261]

[0262] Among them,

[0263] d 22 = 0.8612 + 36.3|ν i | + 0.805|r i |, d23 =7.25+0.874|ν i |+3.45|r i |,

[0264] d 32 =0.0313+3.96|ν i |+0.13|r i |,d 33 =1.9-0.08|ν i |+0.75|r i |. The initial condition of the surface ship is defined as η1(0) = [-4, 15, π / 3] T ,η2(0)=[3,6,-π / 2] T ,η3(0)=[5,-2,π / 3] T ,η4(0)=[3,3,π / 2] T The leader's reference trajectory is chosen as η d =[60 / 100t,60 / 100t,0] T . External disturbance d i Consider [2sin(0.1t),1.5sin(0.2t),sin(0.05t)] T The fault parameter is selected as τ ib =[0.3,0.4,0.2] T The variable parameter of the disturbance observer gain is:

[0265] γ1=2, γ2=6, γ3=6, λ1=2, λ2=6, λ3=0.1, λ4=2, k1=0.5, k2=1.2,

[0266] k3=1.2, k4=0.5, φ1=0.6, δ=0.01, θ=1.2, α1=0.5, α2=6, α3=1, φ=0.001, a=0.5,

[0267] b=0.5, c=1 / 3, m3=35, n3=33, p3=9, q3=15.

[0268] One leader (MSV0) and four followers (MSV1, MSV2, MSV3, MSV4) are considered in the simulation. Faults and input saturation are considered in the test. The simulation results of the proposed control method are shown in Figures 2 to 5 As shown. Figure 2 As shown, notice that the MSV eventually forms a parallelogram and traces the desired trajectory in constant time; Figure 3 (a) Figure 3 (b) Figure 3(c) indicates that the tracking error converges to 0; Figure 4 (a) Figure 4 (b) Figure 4 As can be seen from (c), the proposed disturbance observer can observe the disturbance in a fixed time. The control input is Figure 5 (a) Figure 5 (b) Figure 5 As shown in (c), the fixed-time formation tracking performance is guaranteed, and the proposed control strategy is verified in simulation.

[0269] Comparison of control performance under different controllers:

[0270] The present invention conducts simulation comparisons to demonstrate the improved performance of the proposed control method, taking into account unknown disturbances, faults and saturation constraints.

[0271] The fast fixed-time non-singular terminal sliding mode controller (NTSMC) proposed in this paper is designed as follows:

[0272]

[0273] The sliding mode controller is designed based on ERL. Therefore, the exponential reaching law based controller (ERLBC) is obtained as follows:

[0274]

[0275] The sliding mode controller is designed based on CRL. Then, the constant reaching law based controller (CRLBC) is designed as:

[0276]

[0277] The sliding mode controller is designed based on PRL. The power reaching law based controller (PRLBC) is defined as follows:

[0278]

[0279] The sliding mode controller is designed based on DPRL. The dual power reaching law based controller (DPRLBC) is defined as follows:

[0280]

[0281] Four designed controllers were applied and the error comparison results were as follows: Figure 6 、 Figure 7 As shown in Figure 2, all control methods ensure that the tracking error converges to 0. The comparison of tracking errors can prove that the tracking performance of the proposed SMFC method is improved. The integrated absolute error (IAE) is used to describe the transient and steady-state performance. IAE is defined as Where j = 1, 2, 3. Compared with other methods, it can evaluate the tracking advantage of the proposed SMFC. The control inputs of MSV1, MSV2, MSV3, and MSV4 are plotted in Figure 8 、 Figure 9 、 Figure 10 、 Figure 11 It can be seen that the designed controller not only improves the convergence speed but also reduces chattering. In addition, the simulation results show that the time required to reach zero under uncertainty is guaranteed to be a fixed time, and the upper limit convergence time is related to the design parameters.

[0282] In summary, the simulation results show that the proposed SMFC has a faster convergence speed than other methods. In addition, under uncertain disturbances, the proposed control method can estimate the uncertainty and the system can achieve fast fixed-time convergence performance.

[0283] This paper designs a novel formation controller to achieve fixed-time trajectory tracking for multiple mobile vehicles (MSVs) under model uncertainty, actuator failures, external disturbances, and input saturation constraints. Specifically, an improved sliding mode reaching law is designed by introducing a saturation function and an exponential function. Compared with existing methods, the proposed sliding mode control method achieves faster convergence. Numerical simulations verify the effectiveness of the proposed method.

[0284] This paper studies the fixed-time formation tracking control problem of multiple MSVs subject to model uncertainty, external disturbances, input saturation constraints, and actuator failures. First, a fixed-time disturbance observer is designed to improve the anti-disturbance capability of the multi-MSV formation system. Furthermore, a novel non-singular fast terminal sliding surface is developed to improve control accuracy. Subsequently, an advanced sliding mode reaching law is designed to increase the convergence speed and mitigate the chattering phenomenon caused by sliding mode control by introducing exponential and saturation functions. Compared with finite-time control methods for MSVs, the convergence time of this method is independent of the initial conditions. Finally, simulation results show that this method has the advantages of fast convergence speed and small steady-state error compared to the latest control methods.

[0285] In the technical solution provided by the present invention, the method includes constructing a dynamic model of an ocean surface ship MSV; designing a fixed-time disturbance observer based on the dynamic model of the MSV; designing a fixed-time sliding mode formation controller through the fixed-time disturbance observer and an improved sliding mode convergence law. The method improves the anti-disturbance capability of multiple ocean surface ships through the fixed-time disturbance observer; improves the control accuracy through the fixed-time sliding mode formation controller, and improves the convergence speed and reduces the steady-state error through the improved sliding mode convergence law.

[0286] Each step of the embodiment of the present invention may be performed by an electronic device, including but not limited to a mobile phone, a tablet computer, a portable PC, a desktop computer, etc.

[0287] An embodiment of the present invention provides a computer-readable storage medium, which includes a stored program, wherein when the program is running, the electronic device where the computer-readable storage medium is located is controlled to execute an embodiment of the method for fixed-time sliding mode formation tracking control of multiple ocean surface ships.

[0288] Figure 12 A schematic diagram of an electronic device provided by an embodiment of the present invention is shown in FIG. Figure 12 As shown, the electronic device 21 includes: a processor 211, a memory 212, and a computer program 213 stored in the memory 212 and executable on the processor 211. When the computer program 213 is executed by the processor 211, the method for fixed-time sliding mode formation tracking control of multiple ocean surface ships in the embodiment is implemented. To avoid repetition, they are not described here one by one.

[0289] The electronic device 21 includes, but is not limited to, a processor 211 and a memory 212. Those skilled in the art will understand that Figure 12 It is only an example of the electronic device 21 and does not constitute a limitation of the electronic device 21. It may include more or fewer components than shown in the figure, or a combination of certain components, or different components. For example, the electronic device may also include input and output devices, network access devices, buses, etc.

[0290] The processor 211 may be a central processing unit (CPU), or other general-purpose processors, digital signal processors (DSP), application-specific integrated circuits (ASIC), field-programmable gate arrays (FPGA), or other programmable logic devices, discrete gate or transistor logic devices, discrete hardware components, etc. A general-purpose processor may be a microprocessor or any conventional processor.

[0291] The memory 212 can be an internal storage unit of the electronic device 21, such as a hard disk or memory of the electronic device 21. The memory 212 can also be an external storage device of the electronic device 21, such as a plug-in hard disk, a smart media card (SMC), a secure digital (SD) card, a flash memory card, etc. equipped on the electronic device 21. Furthermore, the memory 212 can also include both an internal storage unit of the electronic device 21 and an external storage device. The memory 212 is used to store computer programs and other programs and data required by the network device. The memory 212 can also be used to temporarily store data that has been output or is about to be output.

[0292] Those skilled in the art will clearly understand that, for the convenience and brevity of description, the specific working processes of the systems, devices and units described above can refer to the corresponding processes in the aforementioned method embodiments and will not be repeated here.

[0293] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A method for tracking and controlling a fixed-time sliding mode formation of multiple ocean surface ships, characterized in that: The method comprises: Step 1: Construct a dynamic model of the ocean surface ship MSV; Step 2: Design a fixed-time disturbance observer based on the dynamic model of MSV; Step 3: Design a fixed-time sliding mode formation controller SMFC through a fixed-time disturbance observer and an improved sliding mode reaching law.

2. The method according to claim 1, characterized in that The step 1 comprises: Assuming there is a group of ocean surface ships MSV, the dynamic model of ocean surface ship i is expressed as: Among them, i=1,2...n, η i =(x i ,y i ,ψ i ) T and v i =(u i ,v i ,r i ) T Respectively represent the position and velocity vectors in the fixed earth coordinate system; M i ∈R 3 , C i (v i )∈R 3 and D i (v i )∈R 3 Represent three dynamic matrices respectively; τ wi =(τ wiu ,τ wiv ,τ wir ) T and τ i =(τ iu ,τ iv ,τ ir ) T Represent external disturbance and input respectively; rotation matrix R i (ψ i ) is expressed as: Considering the problems of actuator failure and input saturation constraints, the control input τ i The definition is as follows: t i =h(τ A )+[(Z(t)-I)h(τ A )+τ ib ]=h(τ A )+τ fi (5) Among them, τ A represents the actual control input, τ ib represents an additional bias fault; Z(t) = diag{z1,z2,z3} represents the coefficient matrix of the actuator; if z1 = 1, z2 = 1, z3 = 1 and τ ib =0, then the i-th actuator has no fault; 0 <z i <1(i=1,2,3) and τ ib ≠0 means the actuator will encounter failure and saturation problems; h(τ A )=sgn(τ A )min(|τ A |,τ AM ) represents the saturation function, where τ AM is a constant related to the actuator limit; Through formula (1) to formula (5), the dynamic system is obtained as follows: Among them, M aci =S i (ψ i )M i S i (ψ i ) T ,C aci =S i (ψ i )(C i (v i )-M i A(r))S i (ψ i ) T ,t aci =S i (ψ i )h(t A ),D aci =S i (ψ i )D i (v i )S i (ψ i ) T ; The matrix of the system is defined as follows: in, Represent the standard parameter matrix, M0, C0, D0 represent the unknown parameter matrix; the dynamic system is further obtained as: in, Indicates that the system is complex and uncertain; The expected trajectory is defined as: Among them, η d ,v d ,u d represents the position, velocity vector and control input of the leading MSV; Position and velocity tracking will be achieved, denoted as η i →η d and v i →v d ; Define relative formation shape h i ∈R 3 , then the formation tracking error is designed as: e 1i =the i -h i -or d (9) Among them, e 1i =[x e1i ,y e1i ,ψ e1i ] represents position error; Therefore, formula (1) can be rewritten as: in, and denote the conversion control input and lumped uncertainty, respectively.

3. The method according to claim 1, characterized in that The step 2 includes: The uncertainty of MSV is considered as a composite perturbation d i , the composite disturbance is estimated by designing a fixed-time disturbance observer; The disturbance observer is designed as: With 2i =γ2z 1i (20) in, Indicates d i Estimates of z 1i ∈R 3 represents an auxiliary state vector, z 2i represents output, γ1>0, γ2>0 represent constant parameters; Since the auxiliary state cannot be measured, z 1i The estimated design is: in, represents the estimation error of the auxiliary state, and γ3>0,λ1>0,λ2>0,0 <k1<1,k2> 1; Assumption 1: Lumped uncertainty d i satisfy is a bounded constant; Lemma 1: Consider a Lyapunov function V(x) that is defined in a neighborhood D of the origin and satisfies Where λ1,λ2>0,0<α<1,β≥1, then the origin of the MSVs is fixed-time stable; therefore, the residence time satisfies Lemma 2: If Assumption 1 holds, and the parameters satisfy γ3>0,λ1>0,λ2>0,0 <k1<1,k2> 1, then the observation error converges to 0 within a fixed time T0, satisfying:

4. The method according to claim 3, characterized in that include: First, we show that the estimation error of the auxiliary state converges to 0 in constant time. Then, we prove the constant-time convergence of the estimation error of the disturbance observer. Step 21. Consider the Lyapunov function as: The derivative function of Lyapunov is: According to Lemma 1, the auxiliary state estimation error converges to 0, and the fixed time is obtained: Step 22: Define the disturbance observation error as According to formula (20), we get: Therefore, combining formula (20) and formula (26), the observation error is further obtained: Therefore, the perturbation observation error will converge to 0 at t>T0, then the fixed-time disturbance observer can accurately estimate the composite uncertainty.

5. The method according to claim 1, wherein The step 3 comprises: Design a fixed-time non-singular sliding mode surface, which is expressed as: The derivative of a non-singular sliding surface is expressed as: Combining the fixed-time disturbance observer, the non-singular sliding mode surface and the improved sliding mode reaching law, the fixed-time sliding mode formation controller SMFC is designed as follows: in i =in i1 +in i2 (31) 6. The method according to claim 3, characterized in that include: Theorem 1: If Assumption 1 holds, and the gain satisfies Non-singular sliding mode control can achieve fixed-time formation tracking, and the stability time T is bounded by: T≤max(T1,T2)+T0 (35) Proof: The proof process is divided into two parts: Part 1: Based on the formula and u i =u i1 +u i2 , the derivative of the non-singular sliding mode surface can be rewritten as: The Lyapunov function is chosen as: The derivative of the Lyapunov function is: Due to the characteristics of the saturation function, two cases are considered: Case 1: ||S i ||>φ; Case 2: ||S i ||<φ; In summary, we get: According to Lemma 1, the fixed time of the reaching capacity of the sliding mode reaching law is obtained as follows: Part 2: When S i When it converges to 0, based on the above results, the convergence of the tracking error will be proved; in order to evaluate the stability of the system, the Lyapunov function is selected as The derivative of the Lyapunov function is: According to Lemma 1, the tracking error will converge to 0 within a fixed time T2, which is: According to part 2, the tracking error converges to 0 after T2; considering part 1, the total convergence time is limited by T3, satisfying T3≤max(T1,T2)+T0; according to If e 1i can converge to 0, then e 2i It will converge to 0 within a fixed time, that is, the fixed-time sliding mode formation controller will achieve formation tracking.

7. The method according to claim 1, characterized in that The improved sliding mode reaching law includes: The sliding mode reaching law is improved based on the saturation function and exponential function as follows: where, α1>0, α2>0, α3 = k / Ω(S i ), Ω(S i ) = η + γ(1 + |S i | / μ) -1 , k>0, η>0, 0<γ<1, μ = α|S0|, 0<Φ = φ c <1, 0<φ<1, is an odd function; in addition, a, b, c are three parameters of the improved sliding mode reaching law, satisfying 0 < c < a < 1 and 0 < b < 1; When the system state is far away from the sliding surface, |S i |>φ, which means the corresponding sliding mode reaching law is: When the system state approaches the sliding surface, It gradually becomes smaller, which has the effect of reducing vibration; When the system state gradually approaches the sliding surface, |S i |≤φ, which means the corresponding reaching law is expressed as:

8. A computer-readable storage medium, characterized in that The computer-readable storage medium includes a stored program, wherein when the program is run, the computer-readable storage medium is controlled to execute the method for fixed-time sliding mode formation tracking control of multiple ocean surface ships according to any one of claims 1 to 7.

9. An electronic device, characterized in that: include: one or more processors; Memory; and one or more computer programs, wherein the one or more computer programs are stored in the memory, and the one or more computer programs include instructions that, when executed by the device, enable the device to perform the method for fixed-time sliding mode formation tracking control of multiple ocean surface ships as described in any one of claims 1 to 7.

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